Nonuniversality of weighted random graphs with infinite variance degree

We prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n , as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).

[1]  Remco van der Hofstad,et al.  Distances in Random Graphs with Finite Mean and Infinite Variance Degrees , 2005, math/0502581.

[2]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[3]  Réka Albert,et al.  Structural vulnerability of the North American power grid. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  P. L. Davies.  The simple branching process : a note on convergence when the mean is infinite , 1978 .

[5]  Remco van der Hofstad,et al.  Degree-Degree Dependencies in Random Graphs with Heavy-Tailed Degrees , 2014, Internet Math..

[6]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .

[7]  Remco van der Hofstad,et al.  Universality for first passage percolation on sparse random graphs , 2012, 1210.6839.

[8]  Omid Amini,et al.  On explosions in heavy-tailed branching random walks , 2011, 1102.0950.

[9]  S. Bornholdt,et al.  Handbook of Graphs and Networks , 2012 .

[10]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[11]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[12]  D. R. Grey.  Explosiveness of age-dependent branching processes , 1974 .

[13]  Béla Bollobás,et al.  Random Graphs , 1985 .

[14]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[15]  G. Hooghiemstra,et al.  First passage percolation on random graphs with finite mean degrees , 2009, 0903.5136.

[16]  Svante Janson,et al.  On percolation in random graphs with given vertex degrees , 2008, 0804.1656.

[17]  Remco van der Hofstad,et al.  Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds , 2014, 1408.0475.

[18]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[19]  A. Barabasi,et al.  Scale-free characteristics of random networks: the topology of the world-wide web , 2000 .

[20]  C. K. Michael Tse,et al.  Analysis of Communication Network Performance From a Complex Network Perspective , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[21]  Manfred Gilli,et al.  Understanding complex systems , 1981, Autom..

[22]  Michael T. Gastner,et al.  The complex network of global cargo ship movements , 2010, Journal of The Royal Society Interface.

[23]  Svante Janson,et al.  A new approach to the giant component problem , 2009 .

[24]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[25]  Julia Komjathy,et al.  Explosive Crump-Mode-Jagers branching processes , 2016, 1602.01657.

[26]  Remco van der Hofstad,et al.  Random Graphs and Complex Networks , 2016, Cambridge Series in Statistical and Probabilistic Mathematics.

[27]  Remco van der Hofstad,et al.  The winner takes it all , 2013, 1306.6467.

[28]  Eliot Marshall,et al.  Complex systems and networks. Connections. Introduction. , 2009, Science.

[29]  Nikolaos Fountoulakis,et al.  Percolation on Sparse Random Graphs with Given Degree Sequence , 2007, Internet Math..

[30]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.